New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions

By Ali H. Bhrawy, Y. Alhamed, Dumitru Băleanu, A. Alzahrani. 2014

110 citations recorded by the source corpus.

Fractional-order generalized Laguerre functions (FGLFs) are proposed depends on the definition of generalized Laguerre polynomials. In addition, we derive a new formula expressing explicitly any Caputo fractional-order derivatives of FGLFs in terms of FGLFs themselves. We also propose a fractional-order generalized Laguerre tau technique in conjunction with the derived fractional-order derivative formula of FGLFs for solving Caputo type fractional differential equations (FDEs) of order ν (0 < ν < 1). The fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on FGLFs and compare them with other methods. Several numerical example are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

Subfields: Control and Systems Engineering, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/39dfac8b2908a6d41b6fc60f05e52724585c1dca

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In Control and Systems Engineering

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Citations in the corpus

The Müntz-Legendre Tau method for fractional differential equations (2016, background): “Recently, spectral methods have received considerable attention to solve fractional differential equations[2-5,11, 13-15, 19-20,26,28-31].”
A Convergent Algorithm for Solving Higher-Order Nonlinear Fractional Boundary Value Problems (2015, methodology): “These techniques include the Adomian’s decomposition method [33], the homotopy perturbation method [6, 3, 42], the variation iteration method [26], the variational iteration method considering Adomian’s polynomials approximation for nonlinear terms [23], the fractional differential transform method ”
Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation (2014, background): “Recently, several numerical techniques are investigated for achieving high accurate solutions for classical fractional differential equations (see for instance [10–18]).”
A review of operational matrices and spectral techniques for fractional calculus (2015, methodology): “We use the fractional-order generalized Laguerre collocation (FGLC) method (see, [97]) to numerically solve the general form of systems of nonlinear FDE, namely Di ui (x) = fi ( x, u1(x), u2(x), .”